We used a target's continuum-component mask to extract the 1D spectrum within the galaxy's stellar-component scale and to define the inner radius used to adjust the \(\mbox{Ly$\alpha $}\)  NB (Section REF ). The continuum-component masks were created with the HST segmentation maps in a similar manner to those used to create object masks for HST prior extractions in [1]}. The HST segmentation map indicates areas in which galaxies are detected and defines the boundaries of the objects (more details are given in Appendix , see also Figures REF  and REF ). If a pixel does not belong to the target on the cutout segmentation map, we replaced the corresponding pixel of the \(F775W\)  cutout with zero and provided an HST target image. We convolved the HST target image with the MUSE PSF at the \(\mbox{Ly$\alpha $}\)  wavelength. Then, we resampled the PSF-convolved HST target image with the spatial resolution of MUSE, aligning it with the MUSE sky coordinate. We normalized the PSF-convolved image so that the peak became 1. To provide the target's continuum-component mask for each source, we applied a threshold value of 0.2 to the normalized convolved image (see Figures REF  and REF ). This threshold was used to extract the spectra from non-AO MUSE cubes for the HST-prior sources in the catalog in [1]}. For the MXDF data, the target's continuum-component masks typically include 66% of the total fluxes in the target's continuum-component images. We defined an inner radius, \(r_{\rm in}\) , beyond which the normalized PSF-convolved profile is below 0.2. We used \(r_{\rm in}\)  in the SB profile measurements for the \(\mbox{Ly$\alpha $}\)  halo tests later, to exclude the SB inside the galaxy's stellar-component scale. The \(r_{\rm in}\)  ranges from 0.8 arcsec to 1.0 arcsec with an average value of 0.81 arcsec (in physical scales at different redshifts, 5.5 to 7.0 kpc with an average of 6.2 kpc).
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